Frustration in the Lords at the lack of progress over the Leveson report has led four peers to table measures to introduce a low-cost arbitration service for defamation, as recommended by Lord Justice Leveson in the defamation bill.

The proposal is one of the central planks of the Leveson report and follows what appear to be fruitless behind-closed-doors cross-party talks on how to respond to the report.

The defamation bill is in front of the Lords on Tuesday and the motion is being supported by four peers including Lady Thatcher's lord chancellor Lord Mackay and Lady Boothroyd, a former Speaker. The other two signatories are Lord Puttnam, the Labour peer, and Lady Scotland, a former Labour attorney general.

The amendments are now gathering support from Lib Dem, Labour and Tory backbenchers. The peers would prefer the government to table its own amendments with the same effect, and at the very least want to hear something on the record from the Lib Dem leader in the Lords, Lord McNally, giving some assurances as to what the government is doing to implement the Leveson report. One source said nothing had been said on the record about progress since Leveson was published two months ago, with government figures briefing one thing to Hacked Off campaigners and another to the newspaper industry.

The Cabinet Office minister Oliver Letwin, who is leading the all-party talks on Leveson, suggested at one point he would produce proposals on a royal charter to oversee a newspaper regulatory body last week, but has failed to do so.

Puttnam has told peers in a briefing note that his amendment is designed to address concerns that access to justice will be unavailable to ordinary citizens in cases of defamation by newspapers because of cost.

He warns that without his proposed amendments "we would be in danger of passing a bill that could only be used in court by the very wealthy".

The amendments require the newspaper industry to set up an arbitration service that would be recognised by the courts.

The purpose of the arbitration service would be to avoid expensive legal actions through the courts and allow access to justice for everybody in cases of defamation, irrespective of their means.

It was intended that the Legal Aid, Sentencing and Punishment of Offenders Act 2012 would in effect abolish "no win, no fee" actions for defamation claimants in their current form from April 2013. But in December the government announced that the act's provisions relating to such actions would not apply to defamation and privacy cases until it was able to introduce other costs protection measures for claimants in such cases, as recommended in the Leveson report.

In his briefing note Puttnam points out: "During the passage of that act, the family of Milly Dowler asked us not to do so. In a letter to the prime minister, they said: 'We could not have [reached a settlement with News International] without a no win, no fee agreement.'

Puttnam continues: "We ignored the Dowler family then; we cannot ignore them again."

Puttnam concedes the government has asked the Civil Justice Council to review costs in defamation cases, but he argues Leveson has provided a ready-made and well-considered solution.

The first amendment inserts a new clause to allows the courts to take into account the use or not of a recognised arbitration service provided by the newspapers' own self-regulatory body (which is called an independent regulatory board in this amendment) when awarding costs and damages in cases of defamation and related civil legal claims.

It also requires the former lord chief justice to establish a body (called a recognition committee in this amendment) to certify that the newspapers have established their independent regulatory board and arbitration service such that the courts can have confidence in their actions and services.

The use of an arbitration service provided by an independent regulatory board would be entirely voluntary. Both the board and the arbitration service would be parts of the newspaper industry's own self-regulatory system. However, with these amendments, courts may vary costs and damages accordingly if such an arbitration service had not been used.

Lord Hunt of Wirral, chairman of the Press Complaints Commission, is holding talks with the newspaper industry to decide how to respond to the amendments.

• This article was amended on 7 February 2013 because the original said The Legal Aid, Sentencing and Punishment of Offenders Act 2012 in effect abolished "no win, no fee" actions for defamation claimants.